A nonlinear transient reaction-diffusion problem from electroanalytical chemistry

A nonlinear reaction-diffusion partial differential equation occurring in models of transient controlled-potential experiments in electroanalytical chemistry is investigated analytically and numerically, with a view to determining a relationship between the concentration of a chemical species and its flux at a reacting electrode. It is shown that a previously known relation that holds for the steady-state case can be used as the first term in a singular perturbation expansion for the time-dependent case. However, in trying to determine the second term, so as to extend the range of validity of the solution, it is found that a phenomenon akin to switchbacking occurs, with the asymptotic details being strongly dependent on the reaction order; this appears to be a consequence of the spatial algebraic decay of the leading-order solution far from the electrode. Comparison of asymptotic results with numerical solutions obtained using finite element methods indicates a relation involving the homogeoneous reaction order for which the two-term asymptotic approximation would work best for all time. Links to problems that involve algebraically decaying boundary layers are briefly discussed.